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Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras

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Abstract

We introduce a class of hyperplane arrangements \(\mathcal {A}\) in \({\mathbb {C}}^n\) that generalise the locus configurations of Chalykh, Feigin and Veselov. To such an arrangement we associate a second order partial differential operator of Calogero–Moser type and prove that this operator is completely integrable (in the sense that its centraliser in \(\mathcal {D}({\mathbb {C}}^n\!\setminus \!\mathcal {A})\) contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras.

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Notes

  1. The operators (1.4) arise in the context of so-called \(\Omega \)-deformation of supersymmetric gauge theories (see, e.g., [N, NW]). The parameters \( \epsilon _1, \epsilon _2, \epsilon _3\) correspond to the ‘\(\Omega \)-deformed’ \(\mathbb {C}^3\) (denoted \( \mathbb {C}_{\epsilon _1} \times \mathbb {C}_{\epsilon _2} \times \mathbb {C}_{\epsilon _3} \) in [GR]), with relation \( \epsilon _1 + \epsilon _2 + \epsilon _3 = 0 \) reflecting the Calabi-Yau condition (see loc. cit., Sect. 1.4).

  2. For technical reasons, it will be more convenient for us to work with a twisted (fractional) ideal which is obtained by replacing \(Q_{\textrm{reg}} = \mathcal {O}(X_{\textrm{reg}}) \) in the above construction by a rank one torsion-free \( \mathcal {O}(X_{\textrm{reg}}) \)-module \( U_{\mathcal {A}} \) (see Sect. 5.1).

  3. Just as the function \( \deg _{L_0} \) on B, its extension to A depends on the ad-nilpotent element \(L_0\). To distinguish between these two degree functions we suppress the dependence of ‘\( \deg \)’ on \( L_0\) in our notation.

  4. Recall, for a (left and/or right) Noetherian domain B, the set \( S = B \! \setminus \!\{0\} \) of all nonzero elements of B satisfies a (left and/or right) Ore condition (Goldie’s Theorem), and the quotient skew-field \( \textbf{Q}(B) \) is obtained in this case by Ore localisation \(B[S^{-1}]\).

  5. The polynomial (5.1) should not be confused with the discriminant of the Coxeter group W, i.e. \(\, \prod _{\alpha \in R_+}(\alpha , x) \,\), which is also denoted frequently by \(\delta \) in the literature.

  6. We should also mention the recent papers [BCES, ER, BZ], where the rings of quasi-invariants are shown to be Cohen-Macaulay in some non-Coxeter cases.

  7. If \( \dim (V) = 1 \), every fat ideal is very fat, and moreover, every very fat one is automatically projective. Unfortunately, this is not in general true when \( \dim (V) > 1 \): there exist fat ideals which are not very fat, and not every very fat ideal is projective (see [BCM]).

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Acknowledgements

We are grateful to P. Etingof, M. Feigin, A. Sergeev and A. Veselov for many questions and stimulating discussions. We also want to thank Pavel Etingof for drawing our attention to new examples of deformed Calogero-Moser operators that appeared in [GR] and Davide Gaiotto for interesting correspondence clarifying to us the origin of these examples. We are especially grateful to Misha Feigin who read the first version of this paper and pointed out several inaccuracies and misprints. The work of the first author is partially supported by NSF grant DMS 1702372 and the 2019 Simons Fellowship. The work of the second author was partially supported by EPSRC under grant EP/K004999/1.

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  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Yuri Berest

  2. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Oleg Chalykh

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  1. Yuri Berest
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Berest, Y., Chalykh, O. Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras. Commun. Math. Phys. 400, 133–178 (2023). https://doi.org/10.1007/s00220-022-04595-4

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